What will we learn? What is mathematical morphology? What is mathematical morphology? Fundamental concepts and operations

Transcription

1 What will we learn? What is mathematical morphology and how is it used in image processing? Lecture Slides ME 4060 Machine Vision and Vision-based Control Chapter 13 Morphological image processing By Dr. Debao Zhou 1 What are the main morphological operations and what is the effect of applying them to binary and grayscale images? What is a structuring element and how does it impact the result of a morphological operation? What are some of the most useful morphological image processing algorithms? What is mathematical morphology? Mathematical morphology is a branch of image processing which has been successfully used to provide tools for representing, describing, and analyzing shapes in images. Initially developed by Jean Serra in the early 1980s Named after the branch of biology that deals with the form and structure of animals and plants. What is mathematical morphology? Basic principle: the extraction of geometrical and topological information from an unknown set (an image) through transformations using another, well-defined, set, known as structuring element (SE). In morphological image processing, the design of SEs, their shape and size, is crucial to the success of the morphological operations that use them. Fundamental concepts and operations Basic set operations: Complement Fundamental concepts and operations Basic set operations Difference Translation Reflection 1

2 Fundamental concepts and operations Logical equivalents of set theory operations Intersection ~ logical AND Fundamental concepts and operations Logical equivalents of set theory operations 1 if A (x,y) andb (x,y) are both 1 C (x,y)= 0 otherwise Similarly: Complement ~ logical NOT Union ~ logical OR Difference ~ A AND (NOT B) The structuring element The structuring element (SE) is the basic neighborhood structure associated with morphological image operations. It is usually represented as a small matrix, whose shape and size impact the results of applying a certain morphological operator to an image. Although a structuring element can have any shape, its implementation requires that it be converted to a rectangular array. For each array, the shaded squares correspond to the members of the SE whereas the empty squares are used for padding, only. The structuring element Examples: square cross MATLAB functions (see Examples 13.1 and 13.2): strel getsequence The two fundamental morphological image operations. Dilation: a morphological operation whose effect is to grow or thicken objects in a binary image. The extent and direction of this thickening is controlled by the size and shape of the structuring element. Mathematically: Dilation geometrical interpretation 2

3 Dilation MATLAB example (13.3) a = [ ; ; ; ; ] se1 = strel('square',2) b = imdilate (a,se1) se2 = strel('rectangle', [1 2]) c = imdilate (a,se2) Erosion: a morphological operation whose effect is to shrink or thin objects in a binary image. The direction and extent of this thinning is controlled by the shape and size of the structuring element. Mathematically: Erosion geometrical interpretation Erosion MATLAB example (13.4) a = [ ; ; ; ; ] se1 = strel('square',2) b = imerode (a,se1) se2 = strel('rectangle', [1 2]) c = imerode (a,se2) Erosion and dilation are dual operations Erosion and dilation can be interpreted in terms of whether a SE fits or hits an image (region) Erosion: Dilation: 3

4 Opening: erosion followed by dilation Mathematically: Opening: example or: In MATLAB: imopen (Tutorial 13.1) Opening: geometrical interpretation Closing: dilation followed by erosion Mathematically: In MATLAB: imclose (Tutorial 13.1) Closing: geometrical interpretation Closing: example 4

5 Hit-or-miss (HoM) transform: a combination of morphological operations that uses two structuring elements (B 1 and B 2 ) designed in such a way that the output image will consist of all locations that match the pixels in B 1 (a hit) and that have none of the pixels in B 2 (a miss). Mathematically: HoM: example (Fig ) or: In MATLAB: bwhitmiss (Tutorial 13.1) Morphological filtering Morphological filters are Boolean filters that apply a many-to-one binary (or Boolean) function h within a window W in the binary input image f(x,y), producing at the output an image g(x,y) given by: Morphological filtering Examples of Boolean operations (h): OR: equivalent to a morphological dilation with a square SE of the same size as W. AND: equivalent to a morphological erosion with a square SE of the same size as W. MAJ (majority): the morphological equivalent to a median filter applicable to binary images. Morphological filtering Application: noise removal Morphological algorithms A cookbook approach In MATLAB: bwmorph (see also Tutorial 13.2) Example 13.6: A = imread('circles.png'); B = bwmorph(a,'skel', Inf); C = bwmorph(b,'spur',inf); D = bwmorph(a,'remove'); E = bwmorph(d,'thicken',3); F = bwmorph(e,'thin',3); 5

6 Morphological algorithms Morphological algorithms Operations supported by bwmorph Example 13.6: Boundary extraction Internal: pixels in A that sit at the edge of A. External: pixels outside A that sit immediately next to A. Morphological gradient: combination of internal and external boundaries. Boundary extraction Example 13.7: a = ones(5,12) a (1:2,1)=0 a (1:2,9)=0 a (4:5,5)=0 b = bwperim(a,8) In MATLAB: bwperim Region filling Region filling In MATLAB: imfill 6

7 Extraction and labeling of connected components Extraction and labeling of connected components Iterative procedure, similar to region filling In MATLAB: bwlabel, bwselect, label2rgb Grayscale morphology Many morphological operations originally developed for binary images can be extended to grayscale images. The mathematical formulation of grayscale morphology uses an input image f(x,y) and a structuring element (SE) b(x,y). Structuring elements in grayscale morphology come in two categories: nonflat and flat. In MATLAB: strel Nonflat SEs can be created with the same function used to create flat SEs, but we must also pass a second matrix (containing the height values) as a parameter. Grayscale morphology Dilation The dilation of an image f(x,y) by a flat SE b(x,y) is defined as: where D b is called the domain of b. For a nonflat SE, b N (x,y): Grayscale morphology Erosion The erosion of an image f(x,y) by a flat SE b(x,y) is defined as: Grayscale erosion and dilation Example 13.8 where D b is called the domain of b. For a nonflat SE, b N (x,y): 7

8 Grayscale morphology Opening The opening of an image f(x,y) by SE b(x,y) is given by: Grayscale opening and closing Example 13.9 Closing The closing of an image f(x,y) by SE b(x,y) is given by: Grayscale morphology Top-hat transformation Top-hat and bottom-hat Example Bottom-hat transformation In MATLAB: imtophat and imbothat Hands-on Tutorial 13.1: Binary morphological image processing (page 325) Tutorial 13.2: Basic morphological algorithms (page 330) 8